CN104792462A - Method for calibrating and verifying horizontal testing on revolution solid equator rotational inertia - Google Patents

Abstract

The invention relates to a method for calibrating and verifying the horizontal testing on revolution solid equator rotational inertia. The method specifically comprises the following steps: (1) measuring the rotational inertia by using a torsional pendulum method, namely, clamping a product on a testing tool, exerting instantaneous driving moment onto a swinging rack in a balance state so as to enable the swinging rack, the product testing tool and the product to freely rotate and vibrate around a rotating shaft, measuring and recording the rotation period in the state, and according to two times of the periodic measurement values, calculating the rotation inertia of the product around the rotating shaft of a tester; (2) analyzing the precision of the rotational inertia, namely, in a calculation formula of the rotational inertia, precisely measuring A and J0 in advance, so that the errors are mainly measurement errors at the vibration period T; and (3) measuring the influence of friction moment to the testing precision of the rotational inertia. By adopting the method, the revolution solid equator rotational inertia can be effectively tested and can be conveniently calibrated and verified, the errors can be somehow controlled, the method is convenient to use and can be conveniently selected according to demands.

Description

The demarcation of the horizontal test of a kind of solid of revolution equator moment of inertia and the method for inspection

Technical field

The present invention relates to solid of revolution moment of inertia technical field, be specifically related to demarcation and the method for inspection of the horizontal test of a kind of solid of revolution equator moment of inertia.

Background technology

Suppose two points at the two ends of an object, and 2 company's being aligneds are through object, object is with this line for rotation center, and its each part rotates to when fixing a position is when rotated the same shape, and this is standard solid of revolution.Mass centre is called for short barycenter, refers to material system be considered to mass concentration in this image point.The abbreviation of mass centre.The barycenter of system of particles is the mean place of system of particles mass distribution.

Summary of the invention

The object of the present invention is to provide demarcation and the method for inspection of the horizontal test of a kind of solid of revolution equator moment of inertia, to carry out equator moment of inertia test calibration and inspection for solid of revolution better, improve test calibration and test effect, conveniently use as required.

For achieving the above object, technical solution of the present invention is as follows.

The demarcation of the horizontal test of solid of revolution equator moment of inertia and a method of inspection, its concrete grammar is:

(1) Inertia Based on Torsion Pendulum Method is utilized to measure moment of inertia: to be first fixed on rocker by test fixture, when equilibrium state, as applied an instantaneous driving moment to rocker, rocker and product test frock just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, being called the skin cycle; By product clamping on test fixture, when equilibrium state, one instantaneous driving moment is applied to rocker equally, rocker, product test frock and product together just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, by this Periodic measurements of twice, the size of product around the moment of inertia of tester rotating shaft can be calculated.

Rotation inerttia equipment such as the inertia in order to adapt to less solid of revolution of solid of revolution is measured, and fixture have employed light material as much as possible as aluminium alloy etc.This rotation inerttia equipment is made up of fixture, leading screw, releasing mechanism, guide rail and pedestal.

According to law of rotation, formed by frock, rotating shaft and object under test, Equation of Motion is:

Jφ′+Kφ+M＝0 (1)

In formula, JJ is moment of inertia; K is the coefficient of torsion of spring; M is damping torque; φ is angular displacement.If the impact ignoring damping has:

φ′+ω ²φ＝0 (2)

In formula:

${\mathrm{\ω}}^2=\frac{K}{J};$

Because:

${\mathrm{\ω}}^2=\left(\frac{2\mathrm{\π}}{T}\right)=\frac{K}{J};$

So:

$J=\frac{K}{{4\mathrm{\π}}^2}{T}^2$

Wherein J:

J＝J ₀+J _d＝AT ²(3)

J ₀for the moment of inertia of the system of rocking itself; J _dfor object under test moment of inertia; T is the hunting period of pallet and determinand; So formula (3) can be written as:

$J_d=\frac{K}{{4\mathrm{\π}}^2}{T}^2-J_0={\mathrm{AT}}^2-J_0---\left(4\right)$

In formula: it is a constant, is determined by torsion-bar spring.

Formula (4) is exactly measure the computing formula of moment of inertia, from formula (4), if A and J ₀given, if measure pallet add determinand after T hunting period, just can calculate the moment of inertia J of object under test _d, discuss below and how to measure A and J ₀.

First, moment of inertia testing apparatus is unloaded, measures its hunting period of T ₀, have:

$J_0={\mathrm{AT}}_0^2---\left(5\right)$

Then, moment of inertia testing apparatus is placed a standard body, measure T hunting period ₀, have according to above formula:

$J_s={\mathrm{AT}}_s^2-J_0---\left(6\right)$

Can be obtained by formula (5) and (6):

$A=\frac{J_s}{T_s^2-T_0^2}$

$J_0={\mathrm{AT}}_0^2$

Therefore have:

$J_s=\frac{J_s}{T_s^2-T_0^2}{T}_s^2-{\mathrm{AT}}_0^2=\frac{J_s}{T_s^2-T_0^2}{T}_s^2-\frac{K}{{4\mathrm{\π}}^2}{T}_0^2---\left(7\right)$

In formula: J _sfor the theoretical value of standard body moment of inertia; J ₀for blank panel inertia; T _shunting period is rocked after adding standard body; T ₀for the blank panel cycle.

(2) moment of inertia precision analysis: in moment of inertia computing formula, A and J ₀can accurately measure in advance.The main source of such error is the measuring error of vibration period T.And cycle T source of error has two: one to be time determination error; Two is ignore the error that damping torque causes.

Time determination error can be obtained by error analysis below the impact that moment of inertia is tested, and the computing formula of moment of inertia is, J=AT ², so according to formula of error transmission, δ J=2AT δ T,

$\mathrm{\η}=\frac{\mathrm{\δJ}}{J}=\frac{2\mathrm{AT\δT}}{{\mathrm{AT}}^2}=2\frac{\mathrm{\δT}}{T}---\left(8\right)$

The measuring error of the cycle capture card of tester only has 0.002ms, is not inaccessible, so time determination error can be ignored because time difference method does very high.

Moment of inertia computing formula releases when ignoring damping action, and according to theory of oscillation, considering damping effect to said apparatus, has been cycle stretch-out.If T ' is for there being the cycle of damping

${T^{\′}}^2=T^2[1+{\left(\frac{0.25\mathrm{\β}}{\mathrm{\π}}\right)}^2]$

$\mathrm{\β}=\ln \left(\frac{{\mathrm{\θ}}_n}{{\mathrm{\θ}}_{n+1}}\right)---\left(9\right)$

If $\frac{{\mathrm{\θ}}_{n+1}}{{\mathrm{\θ}}_n}=0.7,$ Then $\frac{T^{\′}-T}{T}=0.08\%$

Actual amplitude attenuation ratio 0.7 is much smaller.So 0.08% is less than on the impact of test period, last measuring accuracy is not had much affect.

(3) moment of friction is on the impact of equator moment of inertia measuring accuracy: during the test of Inertia Based on Torsion Pendulum Method moment of inertia, first test fixture is fixed on rocker, when equilibrium state, as applied an instantaneous driving moment to rocker, rocker and product test frock just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state.By product clamping on test fixture.When equilibrium state, one instantaneous driving moment is applied to rocker equally, rocker, product test frock and product together just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, by this Periodic measurements of twice, the size of product around the moment of inertia of tester rotating shaft can be calculated.

According to law of rotation, formed by frock, rotating shaft and object under test, Equation of Motion is:

$J=\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+\mathrm{K\θ}+M=0---\left(10\right)$

In formula, J is moment of inertia; K is the coefficient of torsion of spring, is here the moment of torsion of torsion bar and the ratio of angular displacement, is a constant, and it is only relevant with torsion bar rigidity; M is damping torque; θ is angular displacement.If the impact ignoring damping has:

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+{\mathrm{\ω}}^2\mathrm{\θ}=0---\left(11\right)$

In formula: ${\mathrm{\ω}}^2=\frac{K}{J};$

Because: ${\mathrm{\ω}}^2={\left(\frac{2\mathrm{\π}}{T}\right)}^2=\frac{K}{J}$

So: $J=\frac{K}{{4\mathrm{\π}}^2}{T}^2$

In formula, J is the moment of inertia J of system of rocking itself ₀with object under test moment of inertia J _dsum.Therefore above formula can be written as:

$J_d=\frac{K}{{4\mathrm{\π}}^2}{T}^2-J_0={\mathrm{AT}}^2-J_0$

In moment of inertia computing formula, A and J ₀can accurately measure in advance.The main source of such error is the measuring error of vibration period T.And cycle T source of error has two: one to be time determination error; Two is ignore the error that damping torque causes.

Time determination error can be obtained by error analysis below the impact that moment of inertia is tested, and the computing formula of moment of inertia is, J=AT ², so according to formula of error transmission, δ J=2AT δ T,

$\mathrm{\η}=\frac{\mathrm{\δJ}}{J}=\frac{2\mathrm{AT\δT}}{{\mathrm{AT}}^2}=2\frac{\mathrm{\δT}}{T}---\left(12\right)$

Moment of inertia computing formula releases when ignoring damping action, and the multifactorial impact of moment of friction audient, as structure, design, processing, lubrication, service condition, load etc., here by problem reduction, moment of friction is relevant to load (quality of determinand), and moment of friction and twisting angular velocity be directly proportional, so Equation of Motion is:

$J_C=\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+C\left(M\right)\frac{\mathrm{d\θ}}{\mathrm{dt}}+\mathrm{K\θ}=0---\left(13\right)$

In formula, J _cfor moment of inertia; C (M) is friction torque coefficient; Other symbol is with identical above.

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+\frac{C\left(M\right)}{J_C}\frac{\mathrm{d\θ}}{\mathrm{dt}}+\frac{K}{J_C}\mathrm{\θ}=0---\left(14\right)$

Order ${{\mathrm{\ω}}_C}^2=\frac{K}{J_C},$ $2\mathrm{\ζ}=\frac{C\left(M\right)}{J_C},$ Substitution above formula obtains:

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+2\mathrm{\ζ}\frac{\mathrm{d\θ}}{\mathrm{dt}}+{{\mathrm{\ω}}_C}^2\mathrm{\θ}=0---\left(15\right)$

Separate above formula:

Above formula is substituted into:

ω ²＝ω _C ²-ζ ²

$\mathrm{\ω}=\sqrt{{{\mathrm{\ω}}_C}^2-{\mathrm{\ζ}}^2}$

$\mathrm{\ω}=\frac{2\mathrm{\π}}{T}=\sqrt{\frac{K}{J_C}-\frac{C^2\left(M\right)}{{{4J}_C}^2}}$

$\frac{{4\mathrm{KJ}}_C-C^2\left(M\right)}{{{4J}_C}^2}=\frac{{4\mathrm{\π}}^2}{T^2}$

16π ²J _C ²-4KT ²J _C+C ²(M)T ²＝0

$J_C=\frac{{4\mathrm{KT}}^2+\sqrt{{16K}^2{T}^4-{64\mathrm{\π}}^2{C}^2\left(M\right)T^2}}{{32\mathrm{\π}}^2}$

$J_C=\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}+\sqrt{\frac{K^2{T}^4}{{64\mathrm{\π}}^4}-\frac{1}{16}\frac{C^2\left(M\right)T^2}{{\mathrm{\π}}^2}}$

$J_C=\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}+\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}\sqrt{1-\frac{{4\mathrm{\π}}^2{C}^2\left(M\right)}{K^2{T}^2}}$

$J_C\≈\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}+\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}(1-\frac{1}{2}\frac{{4\mathrm{\π}}^2{C}^2\left(M\right)}{K^2{T}^2})$

$J_C\≈\frac{{\mathrm{KT}}^2}{{4\mathrm{\π}}^2}-\frac{C^2\left(M\right)}{4K}$

$\mathrm{\η}=\frac{|J_C-J|}{J}=\frac{C^2\left(M\right)}{4\mathrm{KJ}}---\left(17\right)$

As can be seen from the above equation, the quality of determinand is larger, and the impact of moment of friction is larger, and the precision of test is lower, and therefore, we choose the heaviest determinand and carry out error analysis.

This parameter of friction torque coefficient is bad to be obtained by theory calculate, and we can obtain it by binding tests indirectly.Because the decay of tester torsional oscillation amplitude is brought by friction torque coefficient, completely by formula known, the ratio of adjacent amplitude is relevant with friction torque coefficient, and concrete formula is: $\frac{{\mathrm{\θ}}_n}{{\mathrm{\θ}}_{n+1}}=e^{\mathrm{\ζT}}=e^{\frac{C\left(M\right)}{2J}T}.$

This beneficial effect of the invention is: this invention can be tested for solid of revolution equator moment of inertia effectively, and convenient demarcation and inspection, and its error is controlled to some extent, easy to use, be convenient to select as required.

Embodiment

Below in conjunction with specific embodiment, this law is set forth further.

Embodiment

The demarcation of the horizontal test of solid of revolution equator moment of inertia in the present embodiment and the method for inspection, its concrete grammar is:

(1) Inertia Based on Torsion Pendulum Method is utilized to measure moment of inertia: to be first fixed on rocker by test fixture, when equilibrium state, as applied an instantaneous driving moment to rocker, rocker and product test frock just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, being called the skin cycle; By product clamping on test fixture, when equilibrium state, one instantaneous driving moment is applied to rocker equally, rocker, product test frock and product together just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, by this Periodic measurements of twice, the size of product around the moment of inertia of tester rotating shaft can be calculated.

Rotation inerttia equipment such as the inertia in order to adapt to less solid of revolution of solid of revolution is measured, and fixture have employed light material as much as possible as aluminium alloy etc.This rotation inerttia equipment is made up of fixture, leading screw, releasing mechanism, guide rail and pedestal.

According to law of rotation, formed by frock, rotating shaft and object under test, Equation of Motion is:

Jφ′+Kφ+M＝0 (1)

In formula, J is moment of inertia; K is the coefficient of torsion of spring; M is damping torque; φ is angular displacement.If the impact ignoring damping has:

φ′+ω ²φ＝0 (2)

In formula:

${\mathrm{\ω}}^2=\frac{K}{J};$

Because:

${\mathrm{\ω}}^2=\left(\frac{2\mathrm{\π}}{T}\right)=\frac{K}{J};$

So:

$J=\frac{K}{{4\mathrm{\π}}^2}{T}^2$

Wherein J:

J＝J ₀+J _d＝AT ²(3)

J ₀for the moment of inertia of the system of rocking itself; J _dfor object under test moment of inertia; T is the hunting period of pallet and determinand; So formula (3) can be written as:

$J_d=\frac{K}{{4\mathrm{\π}}^2}{T}^2-J_0={\mathrm{AT}}^2-J_0---\left(4\right)$

In formula: it is a constant, is determined by torsion-bar spring.

Formula (4) is exactly measure the computing formula of moment of inertia, from formula (4), if A and J ₀given, if measure pallet add determinand after T hunting period, just can calculate the moment of inertia J of object under test _d, discuss below and how to measure A and J ₀.

First, moment of inertia testing apparatus is unloaded, measures its hunting period of T ₀, have:

$J_0={\mathrm{AT}}_0^2---\left(5\right)$

Then, moment of inertia testing apparatus is placed a standard body, measure T hunting period ₀, have according to above formula:

$J_s={\mathrm{AT}}_s^2-J_0---\left(6\right)$

Can be obtained by formula (5) and (6):

$A=\frac{J_s}{T_s^2-T_0^2}$

$J_0={\mathrm{AT}}_0^2$

Therefore have:

$J_s=\frac{J_s}{T_s^2-T_0^2}{T}_s^2-{\mathrm{AT}}_0^2=\frac{J_s}{T_s^2-T_0^2}{T}_s^2-\frac{K}{{4\mathrm{\π}}^2}{T}_0^2---\left(7\right)$

In formula: J _sfor the theoretical value of standard body moment of inertia; J ₀for blank panel inertia; T _shunting period is rocked after adding standard body; T ₀for the blank panel cycle.

(2) moment of inertia precision analysis: in moment of inertia computing formula, A and J ₀can accurately measure in advance.The main source of such error is the measuring error of vibration period T.And cycle T source of error has two: one to be time determination error; Two is ignore the error that damping torque causes.

Time determination error can be obtained by error analysis below the impact that moment of inertia is tested, and the computing formula of moment of inertia is, J=AT ², so according to formula of error transmission, δ J=2AT δ T,

$\mathrm{\η}=\frac{\mathrm{\δJ}}{J}=\frac{2\mathrm{AT\δT}}{{\mathrm{AT}}^2}=2\frac{\mathrm{\δT}}{T}---\left(8\right)$

The measuring error of the cycle capture card of tester only has 0.002ms, is not inaccessible, so time determination error can be ignored because time difference method does very high.

Moment of inertia computing formula releases when ignoring damping action, and according to theory of oscillation, considering damping effect to said apparatus, has been cycle stretch-out.If T ' is for there being the cycle of damping

${T^{\′}}^2=T^2[1+{\left(\frac{0.25\mathrm{\β}}{\mathrm{\π}}\right)}^2]$

$\mathrm{\β}=\ln \left(\frac{{\mathrm{\θ}}_n}{{\mathrm{\θ}}_{n+1}}\right)---\left(9\right)$

If $\frac{{\mathrm{\θ}}_{n+1}}{{\mathrm{\θ}}_n}=0.7,$ Then $\frac{T^{\′}-T}{T}=0.08\%$

Actual amplitude attenuation ratio 0.7 is much smaller.So 0.08% is less than on the impact of test period, last measuring accuracy is not had much affect.

(3) moment of friction is on the impact of equator moment of inertia measuring accuracy: during the test of Inertia Based on Torsion Pendulum Method moment of inertia, first test fixture is fixed on rocker, when equilibrium state, as applied an instantaneous driving moment to rocker, rocker and product test frock just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state.By product clamping on test fixture.When equilibrium state, one instantaneous driving moment is applied to rocker equally, rocker, product test frock and product together just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, by this Periodic measurements of twice, the size of product around the moment of inertia of tester rotating shaft can be calculated.

According to law of rotation, formed by frock, rotating shaft and object under test, Equation of Motion is:

$J=\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+\mathrm{K\θ}+M=0---\left(10\right)$

In formula, J is moment of inertia; K is the coefficient of torsion of spring, is here the moment of torsion of torsion bar and the ratio of angular displacement, is a constant, and it is only relevant with torsion bar rigidity; M is damping torque; θ is angular displacement.If the impact ignoring damping has:

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+{\mathrm{\ω}}^2\mathrm{\θ}=0---\left(11\right)$

In formula: ${\mathrm{\ω}}^2=\frac{K}{J};$

Because: ${\mathrm{\ω}}^2={\left(\frac{2\mathrm{\π}}{T}\right)}^2=\frac{K}{J}$

So: $J=\frac{K}{{4\mathrm{\π}}^2}{T}^2$

In formula, J is the moment of inertia J of system of rocking itself ₀with object under test moment of inertia J _dsum.Therefore above formula can be written as:

$J_d=\frac{K}{{4\mathrm{\π}}^2}{T}^2-J_0={\mathrm{AT}}^2-J_0$

In moment of inertia computing formula, A and J ₀can accurately measure in advance.The main source of such error is the measuring error of vibration period T.And cycle T source of error has two: one to be time determination error; Two is ignore the error that damping torque causes.

Time determination error can be obtained by error analysis below the impact that moment of inertia is tested, and the computing formula of moment of inertia is, J=AT ², so according to formula of error transmission, δ J=2AT δ T,

$\mathrm{\η}=\frac{\mathrm{\δJ}}{J}=\frac{2\mathrm{AT\δT}}{{\mathrm{AT}}^2}=2\frac{\mathrm{\δT}}{T}---\left(12\right)$

Moment of inertia computing formula releases when ignoring damping action, and the multifactorial impact of moment of friction audient, as structure, design, processing, lubrication, service condition, load etc., here by problem reduction, moment of friction is relevant to load (quality of determinand), and moment of friction and twisting angular velocity be directly proportional, so Equation of Motion is:

$J_C=\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+C\left(M\right)\frac{\mathrm{d\θ}}{\mathrm{dt}}+\mathrm{K\θ}=0---\left(13\right)$

In formula, J _cfor moment of inertia; C (M) is friction torque coefficient; Other symbol is with identical above.

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+\frac{C\left(M\right)}{J_C}\frac{\mathrm{d\θ}}{\mathrm{dt}}+\frac{K}{J_C}\mathrm{\θ}=0---\left(14\right)$

Order ${{\mathrm{\ω}}_C}^2=\frac{K}{J_C},$ $2\mathrm{\ζ}=\frac{C\left(M\right)}{J_C},$ Substitution above formula obtains:

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+2\mathrm{\ζ}\frac{\mathrm{d\θ}}{\mathrm{dt}}+{{\mathrm{\ω}}_C}^2\mathrm{\θ}=0---\left(15\right)$

Separate above formula:

Above formula is substituted into:

ω ²＝ω _C ²-ζ ²

$\mathrm{\ω}=\sqrt{{{\mathrm{\ω}}_C}^2-{\mathrm{\ζ}}^2}$

$\mathrm{\ω}=\frac{2\mathrm{\π}}{T}=\sqrt{\frac{K}{J_C}-\frac{C^2\left(M\right)}{{{4J}_C}^2}}$

$\frac{{4\mathrm{KJ}}_C-C^2\left(M\right)}{{{4J}_C}^2}=\frac{{4\mathrm{\π}}^2}{T^2}$

16π ²J _C ²-4KT ²J _C+C ²(M)T ²＝0

$J_C=\frac{{4\mathrm{KT}}^2+\sqrt{{16K}^2{T}^4-{64\mathrm{\π}}^2{C}^2\left(M\right)T^2}}{{32\mathrm{\π}}^2}$

$J_C=\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}+\sqrt{\frac{K^2{T}^4}{{64\mathrm{\π}}^4}-\frac{1}{16}\frac{C^2\left(M\right)T^2}{{\mathrm{\π}}^2}}$

$J_C=\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}+\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}\sqrt{1-\frac{{4\mathrm{\π}}^2{C}^2\left(M\right)}{K^2{T}^2}}$

$J_C\≈\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}+\frac{{\mathrm{KT}}^2}{{8\mathrm{\π}}^2}(1-\frac{1}{2}\frac{{4\mathrm{\π}}^2{C}^2\left(M\right)}{K^2{T}^2})$

$J_C\≈\frac{{\mathrm{KT}}^2}{{4\mathrm{\π}}^2}-\frac{C^2\left(M\right)}{4K}$

$\mathrm{\η}=\frac{|J_C-J|}{J}=\frac{C^2\left(M\right)}{4\mathrm{KJ}}---\left(17\right)$

As can be seen from the above equation, the quality of determinand is larger, and the impact of moment of friction is larger, and the precision of test is lower, and therefore, we choose the heaviest determinand and carry out error analysis.

This parameter of friction torque coefficient is bad to be obtained by theory calculate, and we can obtain it by binding tests indirectly.Because the decay of tester torsional oscillation amplitude is brought by friction torque coefficient, completely by formula known, the ratio of adjacent amplitude is relevant with friction torque coefficient, and concrete formula is: $\frac{{\mathrm{\θ}}_n}{{\mathrm{\θ}}_{n+1}}=e^{\mathrm{\ζT}}=e^{\frac{C\left(M\right)}{2J}T}.$

Claims (1)

1. the demarcation of the horizontal test of solid of revolution equator moment of inertia and a method of inspection, is characterized in that: its concrete grammar is:

(1) Inertia Based on Torsion Pendulum Method is utilized to measure moment of inertia: to be first fixed on rocker by test fixture, when equilibrium state, as applied an instantaneous driving moment to rocker, rocker and product test frock just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, being called the skin cycle; By product clamping on test fixture, when equilibrium state, one instantaneous driving moment is applied to rocker equally, rocker, product test frock and product together just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, by this Periodic measurements of twice, the size of product around the moment of inertia of tester rotating shaft can be calculated;

Rotation inerttia equipment such as the inertia in order to adapt to less solid of revolution of solid of revolution is measured, and fixture have employed light material as much as possible as aluminium alloy etc.; This rotation inerttia equipment is made up of fixture, leading screw, releasing mechanism, guide rail and pedestal;

According to law of rotation, formed by frock, rotating shaft and object under test, Equation of Motion is:

Jφ′+Kφ+M＝0 (1)

In formula, JJ is moment of inertia; K is the coefficient of torsion of spring; M is damping torque; φ is angular displacement; If the impact ignoring damping has:

φ′+ω ²φ＝0 (2)

In formula:

${\mathrm{\ω}}^2=\frac{K}{J};$

Because:

${\mathrm{\ω}}^2=\left(\frac{2\mathrm{\π}}{T}\right)=\frac{K}{J};$

So:

$J=\frac{K}{4{\mathrm{\π}}^2}{T}^2$

Wherein J:

J＝J ₀+J _d＝AT ²(3)

J ₀for the moment of inertia of the system of rocking itself; J _dfor object under test moment of inertia; T is the hunting period of pallet and determinand; So formula (3) can be written as:

$J_d=\frac{K}{4{\mathrm{\π}}^2}{T}^2-J_0={\mathrm{AT}}^2-J_0---\left(4\right)$

In formula: it is a constant, is determined by torsion-bar spring;

Formula (4) is exactly measure the computing formula of moment of inertia, from formula (4), if A and J ₀given, if measure pallet add determinand after T hunting period, just can calculate the moment of inertia J of object under test _d, discuss below and how to measure A and J ₀;

First, moment of inertia testing apparatus is unloaded, measures its hunting period of T ₀, have:

J ₀＝AT ₀ ²(5)

Then, moment of inertia testing apparatus is placed a standard body, measure T hunting period ₀, have according to above formula:

J _s＝AT _s ²-J ₀(6)

Can be obtained by formula (5) and (6):

$A=\frac{J_s}{{T_s}^2-{T_0}^2}$

J ₀＝AT ₀ ²

Therefore have:

$J_s=\frac{J_s}{{T_s}^2-T_0^2}{T_s}^2-{\mathrm{AT}}_0^2=\frac{J_s}{{T_s}^2-T_0^2}{T_s}^2-\frac{K}{4{\mathrm{\π}}^2}{T}_0^2---\left(7\right)$

In formula: J _sfor the theoretical value of standard body moment of inertia; J ₀for blank panel inertia; T _shunting period is rocked after adding standard body; T ₀for the blank panel cycle;

(2) moment of inertia precision analysis: in moment of inertia computing formula, A and J ₀can accurately measure in advance; The main source of such error is the measuring error of vibration period T; And cycle T source of error has two: one to be time determination error; Two is ignore the error that damping torque causes;

Time determination error can be obtained by error analysis below the impact that moment of inertia is tested, and the computing formula of moment of inertia is, J=AT ², so according to formula of error transmission, δ J=2AT δ T,

$\mathrm{\η}=\frac{\mathrm{\δJ}}{J}=\frac{2\mathrm{AT\δT}}{{\mathrm{AT}}^2}=2\frac{\mathrm{\δT}}{T}---\left(8\right)$

The measuring error of the cycle capture card of tester only has 0.002ms, is not inaccessible, so time determination error can be ignored because time difference method does very high;

Moment of inertia computing formula releases when ignoring damping action, and according to theory of oscillation, considering damping effect to said apparatus, has been cycle stretch-out; If T ' is for there being the cycle of damping

$T^{\′2}=T^2[1+{\left(\frac{0.25\mathrm{\β}}{\mathrm{\π}}\right)}^2]$

$\mathrm{\β}=\ln \left(\frac{{\mathrm{\θ}}_n}{{\mathrm{\θ}}_{n+1}}\right)---\left(9\right)$

If $\frac{{\mathrm{\θ}}_{n+1}}{{\mathrm{\θ}}_n}=0.7,$ Then $\frac{T^{\′}-T}{T}=0.08\%$

Actual amplitude attenuation ratio 0.7 is much smaller; So 0.08% is less than on the impact of test period, last measuring accuracy is not had much affect;

(3) moment of friction is on the impact of equator moment of inertia measuring accuracy: during the test of Inertia Based on Torsion Pendulum Method moment of inertia, first test fixture is fixed on rocker, when equilibrium state, as applied an instantaneous driving moment to rocker, rocker and product test frock just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state; By product clamping on test fixture; When equilibrium state, one instantaneous driving moment is applied to rocker equally, rocker, product test frock and product together just can around the free torsional oscillations of rotating shaft, measure and record torsional oscillation cycle of this state, by this Periodic measurements of twice, the size of product around the moment of inertia of tester rotating shaft can be calculated;

According to law of rotation, formed by frock, rotating shaft and object under test, Equation of Motion is:

$J\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+\mathrm{K\θ}+M=0---\left(10\right)$

In formula, J is moment of inertia; K is the coefficient of torsion of spring, is here the moment of torsion of torsion bar and the ratio of angular displacement, is a constant, and it is only relevant with torsion bar rigidity; M is damping torque; θ is angular displacement; If the impact ignoring damping has:

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+{\mathrm{\ω}}^2\mathrm{\θ}=0---\left(11\right)$

In formula: ${\mathrm{\ω}}^2=\frac{K}{J};$

Because: ${\mathrm{\ω}}^2={\left(\frac{2\mathrm{\π}}{T}\right)}^2=\frac{K}{J}$

So: $J=\frac{K}{4{\mathrm{\π}}^2}{T}^2$

In formula, J is the moment of inertia J of system of rocking itself ₀with object under test moment of inertia J _dsum; Therefore above formula can be written as:

$J_d=\frac{K}{4{\mathrm{\π}}^2}{T}^2-J_0={\mathrm{AT}}^2-J_0$

In moment of inertia computing formula, A and J ₀can accurately measure in advance; The main source of such error is the measuring error of vibration period T; And cycle T source of error has two: one to be time determination error; Two is ignore the error that damping torque causes;

Time determination error can be obtained by error analysis below the impact that moment of inertia is tested, and the computing formula of moment of inertia is, J=AT ², so according to formula of error transmission, δ J=2AT δ T,

$\mathrm{\η}=\frac{\mathrm{\δJ}}{J}=\frac{2\mathrm{AT\δT}}{{\mathrm{AT}}^2}=2\frac{\mathrm{\δT}}{T}---\left(12\right)$

Moment of inertia computing formula releases when ignoring damping action, and the multifactorial impact of moment of friction audient, as structure, design, processing, lubrication, service condition, load etc., here by problem reduction, moment of friction is relevant to load (quality of determinand), and moment of friction and twisting angular velocity be directly proportional, so Equation of Motion is:

$J_C\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+C\left(M\right)\frac{\mathrm{d\θ}}{\mathrm{dt}}+\mathrm{K\θ}=0---\left(13\right)$

In formula, J _cfor moment of inertia; C (M) is friction torque coefficient; Other symbol is with identical above;

$\frac{d^2\mathrm{\θ}}{{\mathrm{dt}}^2}+\frac{C\left(M\right)}{J_C}\frac{\mathrm{d\θ}}{\mathrm{dt}}+\frac{K}{J_C}\mathrm{\θ}=0---\left(14\right)$

Order ${{\mathrm{\ω}}_C}^2=\frac{K}{J_C},$ substitution above formula obtains:

Separate above formula:

Above formula is substituted into:

$\mathrm{\ω}=\frac{2\mathrm{\π}}{T}=\sqrt{\frac{K}{J_C}-\frac{C^2\left(M\right)}{4{J_C}^2}}$

$\frac{4K{J}_C-C^2\left(M\right)}{4{J_C}^2}=\frac{4{\mathrm{\π}}^2}{T^2}$

16π ²J _C ²-4KT ²J _C+C ²(M)T ²＝0

$J_C=\frac{4{\mathrm{KT}}^2+\sqrt{16K^2{T}^4-64{\mathrm{\π}}^2{C}^2\left(M\right)T^2}}{32{\mathrm{\π}}^2}$

$J_C=\frac{{\mathrm{KT}}^2}{8{\mathrm{\π}}^2}+\sqrt{\frac{K^2{T}^4}{64{\mathrm{\π}}^4}-\frac{1}{16}\frac{C^2\left(M\right)T^2}{{\mathrm{\π}}^2}}$

$J_C=\frac{{\mathrm{KT}}^2}{8{\mathrm{\π}}^2}+\frac{{\mathrm{KT}}^2}{8{\mathrm{\π}}^2}\sqrt{1-\frac{4{\mathrm{\π}}^2{C}^2\left(M\right)}{K^2{T}^2}}$

$J_C\≈\frac{{\mathrm{KT}}^2}{8{\mathrm{\π}}^2}+\frac{{\mathrm{KT}}^2}{8{\mathrm{\π}}^2}(1-\frac{1}{2}\frac{4{\mathrm{\π}}^2{C}^2\left(M\right)}{K^2{T}^2})$

$J_C\≈\frac{{\mathrm{KT}}^2}{4{\mathrm{\π}}^2}-\frac{C^2\left(M\right)}{4K}$

$\mathrm{\η}=\frac{|J_C-J|}{J}=\frac{C^2\left(M\right)}{4\mathrm{KJ}}---\left(17\right)$

As can be seen from the above equation, the quality of determinand is larger, and the impact of moment of friction is larger, and the precision of test is lower, and therefore, we choose the heaviest determinand and carry out error analysis;

This parameter of friction torque coefficient is bad to be obtained by theory calculate, and we can obtain it by binding tests indirectly; Because the decay of tester torsional oscillation amplitude is brought by friction torque coefficient, completely by formula known, the ratio of adjacent amplitude is relevant with friction torque coefficient, and concrete formula is: